If math is more than proof, we need to better celebrate the rest of it

The piece proposes and characterizes the notion of a "motivated explanation" in contrast with proof: in a proof definitions sit at the start and every claim must be correct and follow as a necessary implication from what comes before; in a motivated explanation definitions sit in the middle, new constructions enter the vocabulary only once the problem they address has been clearly established, and it is acceptable and often desirable to start from a relatable idea that is not quite right and requires correction. It distills a kind of work already present but not formally named in mathematics (such as the narrative genre Michael Nielsen calls "discovery fiction") into a discussable, evaluable category, and argues its scope is not only why a theorem is true but why the theorem is the right one to pose and how it is used in surrounding context. This is an opinion and concept-definition piece; its basis is a contrastive description of two modes of writing plus the author's own working guideline, "I want this to feel like you could have discovered it yourself," rather than experiments or statistics.

The piece argues that such work deserves academic credit comparable to what generating new proofs of open problems has historically received, and notes that its validity is intrinsically squishier than a proof's because defining human understanding itself is squishier, while "motivated" is a more verifiable property than alternatives like "lucid" or "demystifying" and is enough to serve as a practical measure. It sets the existing advantage of proof as a measure of progress — you can clearly define what does and does not have a proof yet — alongside the reality that understanding resists binary verification, and argues that shying away from more subjective metrics means shying away from the more human aspects of the field. The argument rests on conceptual analysis and analogy (checking whether key ideas are motivated is not unlike checking whether the steps of a proof follow logically), and it explicitly concedes that "there will never be Lean for motivated explanations."

The piece uses several exemplars to show this work has long existed and adds tremendous value: Part IV of the Princeton Companion to Mathematics, whose field introductions by experts offer intuition more typically found at a blackboard; Timothy Gowers' account that editing the book took roughly half his working time for about five years and that he probably would not have been offered the chance without being a Fields Medalist; Bill Thurston's famous essay on proof and progress and his list of "non-credit-producing activities"; and Timothy Chow's A beginner's guide to forcing together with its notion of an "open exposition problem." It threads these scattered examples into a single line of argument: since these contributions are acknowledged as valuable, why should they follow a Fields Medal rather than contribute to it? The basis is specific quoted texts and interview passages cited in the piece, making this a citation-based argument from documents and figures rather than a quantitative assessment.

Using Erdős Problem 1196 (the asymptotic primitive sets conjecture) as a case, the piece argues that the understanding work following an AI-generated proof is likewise worth celebrating: Liam Price submitted a solution arising from interaction with GPT-5.4 Pro, Nat Sothanaphan and Jared Lichtman interpreted the AI's approach and cleaned the proof into human-readable form, and in May a paper by Boris Alexeev, Kevin Barreto, Yanyang Li, Jared Duker Lichtman, Liam Price, Jibran Iqbal Shah, Quanyu Tang and Terence Tao expanded on the key idea, which clarified not only the original problem but many around it, for instance offering a cleaner proof of the Erdős Primitive Set Conjecture. The piece stresses that the value is not that one more Erdős problem could be ticked off as solved, but that our understanding of primitive sets is notably cleaner and more satisfying than at the start of 2026, and that the work expanding, clarifying and contextualizing the key idea arguably deserves more celebration. The basis is the specific events and author list narrated in the piece, a case account; the piece also observes that "every AI-generated proof is born an unsolved exposition problem."

The piece is aimed at policy and culture discussions inside mathematics, and applies to departments, journals, funders and early-career researchers thinking about how to define and reward contributions when proof-generating tools are widespread; its proposals are set in the discipline of mathematics and do not offer a quantitative scheme directly transferable to other fields.

The piece explicitly acknowledges that whether an explanation is motivated is not binary and that a rubric would have to be agreed upon, so how this would work in actual review remains an open question; moreover, the loaded text is the blog page itself and does not include the external papers, interviews and videos it cites, so readers wishing to verify the details and author contributions of the Erdős Problem 1196 paper would still need to consult the original literature.